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Weyl Sum

An exponential sum of the form

sum_(n=1)^Ne^(2piiP(n)),
(1)

where P(n) is a real polynomial (Weyl 1914, 1916; Montgomery 2001). Writing

e(theta)=e^(2piitheta),
(2)

a notation introduced by Vinogradov, Weyl observed that

|sum_(n=1)^(N)e(P(n))|^2=sum_(n=1)^(N)sum_(m=1)^(N)e(P(m)-P(n))
(3)
=sum_(n=1)^(N)sum_(h=1-n)^(N-n)e(P(n+h)-P(n))
(4)
=sum_(h=-N+1)^(N-1)sum_(1<=n<=N; 1-h<=n<=N-h)e(P(n+h)-P(n))
(5)
=N+2R[sum_(h=1)^(N-1)sum_(n=1)^(N-h)e(P(n+h)-P(n))],
(6)

a process known as Weyl differencing (Montgomery 2001).

Weyl was able to use this process to show that if

P(x)=sum_(i=0)^da_ix^i
(7)

is a real polynomial and at least one of a_1, ..., a_d is irrational, then {P(n)} is uniformly distributed (mod 1).

See also

van der Corput's Inequality, Weyl's Criterion

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References

Berry, M. V. and Goldberg, J. "Renormalisation of Curlicues." Nonlinearity 61, 1-26, 1988.Lehmer, D. H. and Lehmer, E. "Picturesque Exponential Sums, I." Amer. Math. Monthly 86, 725-733, 1979.Montgomery, H. L. "Harmonic Analysis as Found in Analytic Number Theory." In Twentieth Century Harmonic Analysis--A Celebration. Proceedings of the NATO Advanced Study Institute Held in Il Ciocco, July 2-15, 2000 (Ed. J. S. Byrnes). Dordrecht, Netherlands: Kluwer, pp. 271-293, 2001.Montgomery, H. L. Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis. Providence, RI: Amer. Math. Soc., 1994.Pickover, C. A. "Is the Fractal Golden Curlicue Cold?" Visual Comput. 11, 309-312, 1995.Stewart, I. Another Fine Math You've Got Me Into.... New York: Freeman, 1992.Weyl, H. "Über ein Problem aus dem Gebiete der diophantischen Approximationen." Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl., 234-244, 1914. Reprinted in Gesammelte Abhandlungen, Band I. Berlin: Springer-Verlag, pp. 487-497, 1968.Weyl, H. "Über die Gleichverteilung von Zahlen mod. Eins." Math. Ann. 77, 313-352, 1916. Reprinted in Gesammelte Abhandlungen, Band I. Berlin: Springer-Verlag, pp. 563-599, 1968. Also reprinted in Selecta Hermann Weyl. Basel, Switzerland: Birkhäuser, pp. 111-147, 1956.

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Weyl Sum

Cite this as:

Weisstein, Eric W. "Weyl Sum." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WeylSum.html

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